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Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

  • Eduardo Reyes
  • Tianqi Wang
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Abstract

We establish several results about Patterson--Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves--Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are C1-regular, from which we deduce that the growth indicator is C1-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim--Oh--Wang. Our methods are dynamical, and we exploit the fact due to Kim--Oh and Blayac--Canary--Zhu--Zimmer that Bowen--Margulis--Sullivan measures for relatively Anosov groups are finite and mixing.

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Cite

BibTeXRIS
BibTeX
@misc{reyes2026regularity,
  title = {{Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups}},
  author = {Eduardo Reyes and Tianqi Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.13730},
  eprint = {2607.13730},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.13730},
}
RIS
TY  - UNPB
TI  - Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups
AU  - Eduardo Reyes
AU  - Tianqi Wang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2607.13730
UR  - https://arxiv.org/abs/2607.13730
AB  - We establish several results about Patterson--Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves--Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are C^1-regular, from which we deduce that the growth indicator is C^1-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim--Oh--Wang. Our methods are dynamical, and we exploit the fact due to Kim--Oh and Blayac--Canary--Zhu--Zimmer that Bowen--Margulis--Sullivan measures for relatively Anosov groups are finite and mixing.
ER  -